What happens if the pilot turns to the right using ailerons but then simultaneously applies left rudder? I'm asking about a large aircraft e.g. Boeing 747-800 or Airbus A380, or any other aircraft. If there are any negative effects, what mechanical or other measures are in place to avoid this (besides strict training)?
what mechanical measures are in place to avoid this?
The short answer is none.
Cross-controlling (the usual term for applying opposing aileron and rudder) is sometimes desirable. Its greatest use in large aircraft is to align an airplane with the runway just before touchdown from a crabbed approach in a crosswind while keeping the upwind wing from rising. In other words, if you had a crosswind from the right and you were crabbed to the right to counteract it, just before touchdown you would apply left rudder to swing the nose left while applying right aileron to keep the right wing from coming up (or maybe put the right wing down a little).
What happens when the pilot makes a right aileron turn but then gives a left rudder turn simultaneously
I'm making the assumption that you're talking about a turn not as part of a crosswind landing, and I suspect that your assumption is that pilots of large aircraft normally use coordinated rudder input when they use the ailerons. However, that is not the case. Once the aircraft is off the ground and absent any need to counter asymmetrical thrust from a failed engine, pilots generally take their feet off the rudder pedals and do not put them back on until short final. Yaw dampers or the equivalent for fly-by-wire systems take care of any adverse yaw ailerons may produce. Thus a cross-control scenario for en route flying would mean the pilot would have to make not one, but two errors. They shouldn't have had their feet on the rudders in the first place. Then they shouldn't have cross controlled.
But let's say that, for whatever reason, during en route flight, a pilot deliberately cross-controls once and then releases the controls. Generally speaking, that would not be disastrous. However, if they held a cross-control condition or kept repeating it in sequence, that's another story and would depend on a number of other conditions, and you could certainly come up with scenarios that ended in the destruction of the aircraft. Although I'm not sure the fly-by-wire airplanes would permit you to do that, but I've never flown any.
If you were in a 747-100/200 en route, you could if you wanted give it left aileron and right rudder in proportions that matched. In other words, the turning force of the ailerons would be cancelled out by the turning force of the rudder. The airplane would keep going straight ahead. However, you would be in a sideslip, and you'd have to increase the wing's angle of attack to keep from descending, which would mean more drag. Plus, there would be all that additional drag from the rudder. In other words, you'd be burning a lot more fuel than you should. Plus, you'll get a call from the purser asking what the hell is going on because gravity is no longer perpendicular to the cabin floor.
The aircraft is now in uncoordinated flight and if you stalled, a spin could occurred.
What is a spin? A spin is autorotation about the vertical axis and a shallow, rotating, downward path.
Some aircraft spin are prohibited because the pilot might not be able to recover and in time the aircraft impacts the ground and can cause total destruction of the aircraft and loss of life of the pilot and passengers.
Believe it or not, in the barn storming days of aviation pilots would be on top of the clouds and not be able to see the ground. They would actually put their airplanes into a spin and begin to descend in the clouds. The plan is to break out of the clouds and see the ground before it is to late to recovery from the spin.
Now, in the modern Era of Aviation, there is electronic navigation to guide us in for a landing. This is called an instrument approached.
In some airplanes uncoordinated flight is used to land in crosswinds. The pilot uses the rudders to keep the nose of the aircraft lined up with the centerline of the runway and the ailerons for wind drift to keep the airplane over the runway. This is called a side slip. If the wind blew the airplane to the left of the runway, the pilot would apply right aileron and vice versa.
Another flight maneuver is called a crab and is also used for crosswind landings. The nose of the airplane is aimed into the wind direction to keep the ground path lined up with the runway. Wind from the left aim, the nose is aimed left and wind from the right, aim right. How much angle you use depend of the wind speed. In the flare, the wind correction angle is taken out and and the pilot lands the airplane.